Evaluate. The differential is exact.
-$$\int _ { ( 0,0,0 ) } ^ { ( 1,1,1 ) } 8 x e ^ { 4 x ^ { 2 } + 9 y ^ { 2 } + 6 z ^ { 2 } } d x + 18 y e ^ { 4 x ^ { 2 } + 9 y ^ { 2 } + 6 z ^ { 2 } } d y + 12 z e ^ { 4 x ^ { 2 } + 9 y ^ { 2 } + 6 z ^ { 2 } } d z$$
A) $e ^ { 4 } + e ^ { 9 } + e ^ { 6 } - 1$
B) $\mathrm { e } ^ { 19 } - 3$
C) $\mathrm { e } ^ { 19 } - 1$
D) 0

Question

Quizplus · Accepted Answer

The given integral is the line integral of a gradient field, which can be evaluated by finding the potential function $f(x, y, z) = e^{4x^2 + 9y^2 + 6z^2}$. The value of the integral is the difference in the potential function evaluated at the endpoints, which is $f(1,1,1) - f(0,0,0) = e^{19} - 1$.

Evaluate the Differential Is Exact A) e4+e9+e6−1e ^ { 4 } + e ^ { 9 } + e ^ { 6 } - 1e4+e9+e6−1

Evaluate the Differential Is Exact A) $e ^ { 4 } + e ^ { 9 } + e ^ { 6 } - 1$